Theorem islaut

Description: The predictate "is a lattice automorphism." (Contributed by NM, 11-May-2012)

	Ref	Expression
Hypotheses	lautset.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	lautset.l	⊢ ≤ = ( le ‘ 𝐾 )
	lautset.i	⊢ 𝐼 = ( LAut ‘ 𝐾 )
Assertion	islaut	⊢ ( 𝐾 ∈ 𝐴 → ( 𝐹 ∈ 𝐼 ↔ ( 𝐹 : 𝐵 –1-1-onto→ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ↔ ( 𝐹 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑦 ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		lautset.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		lautset.l	⊢ ≤ = ( le ‘ 𝐾 )
3		lautset.i	⊢ 𝐼 = ( LAut ‘ 𝐾 )
4	1 2 3	lautset	⊢ ( 𝐾 ∈ 𝐴 → 𝐼 = { 𝑓 ∣ ( 𝑓 : 𝐵 –1-1-onto→ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ↔ ( 𝑓 ‘ 𝑥 ) ≤ ( 𝑓 ‘ 𝑦 ) ) ) } )
5	4	eleq2d	⊢ ( 𝐾 ∈ 𝐴 → ( 𝐹 ∈ 𝐼 ↔ 𝐹 ∈ { 𝑓 ∣ ( 𝑓 : 𝐵 –1-1-onto→ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ↔ ( 𝑓 ‘ 𝑥 ) ≤ ( 𝑓 ‘ 𝑦 ) ) ) } ) )
6		f1of	⊢ ( 𝐹 : 𝐵 –1-1-onto→ 𝐵 → 𝐹 : 𝐵 ⟶ 𝐵 )
7	1	fvexi	⊢ 𝐵 ∈ V
8		fex	⊢ ( ( 𝐹 : 𝐵 ⟶ 𝐵 ∧ 𝐵 ∈ V ) → 𝐹 ∈ V )
9	6 7 8	sylancl	⊢ ( 𝐹 : 𝐵 –1-1-onto→ 𝐵 → 𝐹 ∈ V )
10	9	adantr	⊢ ( ( 𝐹 : 𝐵 –1-1-onto→ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ↔ ( 𝐹 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑦 ) ) ) → 𝐹 ∈ V )
11		f1oeq1	⊢ ( 𝑓 = 𝐹 → ( 𝑓 : 𝐵 –1-1-onto→ 𝐵 ↔ 𝐹 : 𝐵 –1-1-onto→ 𝐵 ) )
12		fveq1	⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) )
13		fveq1	⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) )
14	12 13	breq12d	⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 ‘ 𝑥 ) ≤ ( 𝑓 ‘ 𝑦 ) ↔ ( 𝐹 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑦 ) ) )
15	14	bibi2d	⊢ ( 𝑓 = 𝐹 → ( ( 𝑥 ≤ 𝑦 ↔ ( 𝑓 ‘ 𝑥 ) ≤ ( 𝑓 ‘ 𝑦 ) ) ↔ ( 𝑥 ≤ 𝑦 ↔ ( 𝐹 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑦 ) ) ) )
16	15	2ralbidv	⊢ ( 𝑓 = 𝐹 → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ↔ ( 𝑓 ‘ 𝑥 ) ≤ ( 𝑓 ‘ 𝑦 ) ) ↔ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ↔ ( 𝐹 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑦 ) ) ) )
17	11 16	anbi12d	⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 : 𝐵 –1-1-onto→ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ↔ ( 𝑓 ‘ 𝑥 ) ≤ ( 𝑓 ‘ 𝑦 ) ) ) ↔ ( 𝐹 : 𝐵 –1-1-onto→ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ↔ ( 𝐹 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑦 ) ) ) ) )
18	10 17	elab3	⊢ ( 𝐹 ∈ { 𝑓 ∣ ( 𝑓 : 𝐵 –1-1-onto→ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ↔ ( 𝑓 ‘ 𝑥 ) ≤ ( 𝑓 ‘ 𝑦 ) ) ) } ↔ ( 𝐹 : 𝐵 –1-1-onto→ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ↔ ( 𝐹 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑦 ) ) ) )
19	5 18	bitrdi	⊢ ( 𝐾 ∈ 𝐴 → ( 𝐹 ∈ 𝐼 ↔ ( 𝐹 : 𝐵 –1-1-onto→ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ↔ ( 𝐹 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑦 ) ) ) ) )